Macroscopic quantum coherence and entanglement in mechanical systems

arXiv:2503.08324 · quant-ph · Submitted 2025-03-11 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Macroscopic quantum coherence and entanglement in mechanical systems".

Mira: Here's the gist of the paper. The gist The authors introduce two measures, extensive size and entangled size, to quantify macroscopic quantum properties in mechanical systems, demonstrating evidence for genuine multipartite entanglement of 106 to 107 atoms in recent mechanical superposition states.

Kai: Measures of Macroscopicity The paper introduces two measures that capture independent macroscopic properties: i) extensive size, measuring the degree of coherence in position or momentum observables relative to atomic-scale units, and ii) entangled size, quantifying the number of entangled subsystems.

Mira: These measures unify many desirable features of past proposals while having rigorously justified interpretations from quantum information theory in terms of coherence and multipartite entanglement.

Kai: The measures are based on quantum Fisher information (QFI).

Title and authors: Kai: So we're looking at this paper titled "Macroscopic quantum coherence and entanglement in mechanical systems." It’s about trying to figure out what makes a state truly macroscopic quantum mechanically.

Mira: Right, it’s tackling the problem of how to quantify something that is so hard to pin down, you know, linking the physics of large systems with the limits of quantum mechanics itself.

Kai: The authors introduce two specific measures: extensive size and entangled size. These are meant to capture two different aspects of what we call macroscopicity.

Mira: Exactly. They set out to unify some ideas from previous research, but they’ve grounded these new measures in the language of quantum information theory, specifically using quantum Fisher information or QFI as their basis for everything.

Lev: From my side, I’m interested in how computable these measures are. If we want to run this on actual hardware, we need to know if we can actually extract these values from experimental data without it taking forever.

Kai: Well, the paper shows they can be estimated directly from experimental data using things like micro-mechanical oscillators and diffracting molecules.

Mira: And they provide some really concrete numbers there, suggesting that these states actually exhibit genuine multipartite entanglement in systems with one hundred six to one hundred seven atoms when we look at recent mechanical superposition states.

The paper's summary: Kai: So, the core idea is that there isn't just one single measure that can describe macroscopicity, which is what Leggett suggested earlier. Instead, they propose these two distinct measures to capture independent properties.

Mira: They call the first one extensive size, which measures the degree of coherence in position or momentum observables relative to some atomic scale unit.

Kai: And the second one is entangled size, which quantifies how many entangled subsystems are actually present in a state.

Lev: When I look at this from an error correction standpoint, I’m wondering how these definitions map onto the actual overhead we’d need for an AI to verify these properties on a real quantum system.

Mira: The paper shows that both of these measures have rigorous interpretations from quantum information theory, specifically connecting them to coherence and multipartite entanglement.

Kai: They define extensive size as Next(rho, A) by using the formula F(rho, A) divided by four times A squared zero, where A is an extensive observable.

Mira: And the entangled size is defined as Nent(rho, A Pi) by taking the QFI and dividing it by the sum over all variances of those local observables for a given partition Pi.

The paper's improvements: Lev: I’m looking at how they handle real systems, and it’s interesting that they use these measures to create lower bounds using experimental data from mechanical oscillators and diffracting molecules.

Kai: They do, and the results are quite telling. For example, in diffraction experiments on diffracting molecules, they found an extensive size Next(rho0, Q) greater than or equal to one point four times ten to the power of fourteen for those states <ref:2503.08324#pg3>.

Mira: That number is quite large and suggests a strong level of coherence in position observables relative to atomic scales. It really confirms the expectation that these states behave like true macroscopic spatial superpositions <ref:2503.08324#pg2>, which is significant given how hard it is to maintain coherence at that scale.

Lev: If we try to run this on real hardware, the numbers suggest we’re looking at some very high coherence levels, but I gotta ask if that level of coherence can actually be maintained without decoherence ruining everything.

Kai: They also have these illustrative examples, like Leggett's crystal example showing that for a momentum superposition state, the entangled size Nent(rho1 s, Q, Π) is about ten to the power of thirteen <ref:2503.08324#pg2> <ref:2503.08324#pg1>.

Mira: That’s a really high number for entanglement in that specific context. It shows how these measures can actually be used as an entanglement witness <ref:2503.08324#pg2>, which is a useful property for testing multipartite correlations.

Conclusion: Kai: So, to wrap up this paper on "Macroscopic quantum coherence and entanglement in mechanical systems," the authors show that these two measures are quite correlated when you look at realistic physical cases.

Mira: They also point out a significant gap between what current experiments can actually do and what we might need for quantum gravity tests, which is about twenty orders of magnitude difference in terms of extensive size and five in entangled size.

Lev: From the perspective of running this on hardware, that separation tells me that if we want to test things at the gravity scale using this method, we're talking about a completely different level of coherence than what’s achievable right now.

Kai: They suggest extending the analysis to linear combinations of observables like R(t) equals Q plus tP to explore relativistic generalizations, and they also mention that for superconducting qubits, the entangled size in momentum can reach values comparable to Leggett's original crystal example.

Mira: The overall hope here is that this work leads to a unifying formalism for evaluating quantum macroscopicity across a wide range of experimental systems.

Lev: I just think the practical path forward for me is figuring out how we can build an AI system that actually uses these results to predict decoherence rates based on the relationship between the decoherence rate Gamma and Next(rho, Q) <ref:2503.08324#pg1>.

Kai: That’s a great direction for next steps, moving from just measuring the size to predicting how long that coherence will actually last.

Naturwissenschaftlich-Technische Fakultät, Universität Siegen · Department of Physics, ETH Zürich

quant-ph

Submitted: 2025-03-11

Updated: 2026-10-22

Comments: 31 pages, 4 figures. Comments welcome

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 69/100

The gist: The gist The authors introduce two measures, extensive size and entangled size, to quantify macroscopic quantum properties in mechanical systems, demonstrating evidence for genuine multipartite

Key concepts

Extensive Size: This measure quantifies the degree of coherence in a state with respect to a macroscopic observable, like the center-of-mass position. It is defined by dividing the quantum Fisher information by a term related to the square of that observable's scale, effectively measuring how coherent the system is relative to its size.
Entangled Size: This measure focuses on quantifying multipartite correlations between different parts of a system. It calculates this by taking the quantum Fisher information and dividing it by the sum of variances of local observables across all subsystems in a partition, helping to determine if genuine entanglement exists.
Quantum Fisher Information (QFI): QFI is a fundamental tool derived from quantum mechanics that measures the minimal uncertainty with which a parameter encoded in a state can be estimated. The paper uses QFI as the basis for defining both extensive and entangled sizes, linking them directly to state coherence and entanglement.
Macroscopic Quantum Properties: This refers to large-scale quantum behaviors observed in mechanical systems, such as those involving 106 to 107 atoms. The paper provides rigorous mathematical tools—extensive size and entangled size—to quantify these complex quantum features that go beyond simple atomic-level descriptions.

Terminology

Coherence — 3× in this paper · explained in 2 episodes
Entanglement — 3× in this paper · explained in 4 episodes
Multipartite Entanglement — 2× in this paper · explained in 3 episodes

Summary

The gist The authors introduce two measures, extensive size and entangled size, to quantify macroscopic quantum properties in mechanical systems, demonstrating evidence for genuine multipartite entanglement of 106 to 107 atoms in recent mechanical superposition states.

Measures of Macroscopicity

The paper introduces two measures that capture independent macroscopic properties: i) extensive size, measuring the degree of coherence in position or momentum observables relative to atomic-scale units, and ii) entangled size, quantifying the number of entangled subsystems. These measures unify many desirable features of past proposals while having rigorously justified interpretations from quantum information theory in terms of coherence and multipartite entanglement. The measures are based on quantum Fisher information (QFI).

Extensive Size

The extensive size quantifies the amount of coherence in the state with respect to the eigenbasis of a macroscopic observable. This is defined as Next(ρ, A):= F(ρ, A) / 4A 2 0. For mechanical systems, the authors focus on observables like the center-of-mass position multiplied by total mass. The relationship between Next and Leggett’s extensive difference is shown to be a square relationship, Next(ρ, Q) = Var(ρ, Q) / Q 2 0.

Entangled Size

The entangled size aims to capture multipartite correlations between parts of a system. It is defined as Nent(ρ, A, Π):= F(ρ, A) / (Sum over all variances of the local observables AΠi). The ability of Nent to act as an entanglement witness was proven in Refs. [26, 28]. Key properties include the maximum size bound, Nent(ρ, A, Π) ≤ n, and the independent systems property, which shows that Nent is never increased under combining independent systems.

Illustrative Examples

The paper illustrates these measures using several physical systems:

Leggett’s crystal example shows that for a momentum superposition state, the entangled size Nent(ρ1 s, Q, Π) ≈ N ∼ 10 13 – confirming the expectation that this state resembles a true macroscopic spatial superposition.
For mechanical oscillators in thermal states at temperature T with mean photon number ¯n, the entangled size is estimated as Nent(ρ, Qk, Π) ≈ Nk · (1 / (2¯n + 1)) · ∆X zp k / ∆u squared.
In diffraction experiments on diffracting molecules, the extensive size Next(ρ0, Q) ≳ 1.4 × 10 14 was obtained.

Conclusions and Outlook

The work concludes that the two measures are well correlated in realistic cases. The authors note a notable separation between current experiments and quantum gravity test proposals of roughly 20 orders of magnitude in Next and 5 in Nent. They suggest extending the analysis to linear combinations of observables R(t) = Q + tP to explore relativistic generalizations. Furthermore, the paper suggests that for superconducting qubits, the entangled size in momentum can reach values comparable to Leggett’s original crystal example. The authors hope this work leads to a unifying formalism for evaluating quantum macroscopicity across a wide range of experimental systems.

How it works

The measures are derived from the quantum Fisher information (QFI). The QFI quantifies the minimal uncertainty with which one can estimate a parameter encoded into a probe state. The extensive size is related to the coherence cost of preparing a state, defined as Next(ρ, A):= F(ρ, A) / 4A 2 0.

Estimates from Wigner Function

When applying the measures to cases where QFI is more directly known, such as using Wigner function data from mechanical oscillators, the resulting extensive and entangled sizes are shown in Table I. The largest values are found when the QFI-maximising direction is the position quadrature.

Diffraction Analysis

In diffraction experiments, the relevant QFI can be lower-bounded from detection statistics using Eq. (32). This leads to an extensive size Next(ρ0, Q) ≳ 1.4 × 10 14 for the diffracted molecules. The entangled size estimate Nent(ρ0, Q, Π) ≈ 5 atoms was found in this context.

Oscillator Details

For mechanical oscillators, the mode functions have orthogonality relations. The variance of the local observable Ai is derived as Var(ρ, Ai) = X i ζ(i, k,l) squared Var(ρ, Ql). This leads to the final entangled size estimate Nent(ρ, Qk, Π) ≈ Nk · (1 / (2¯n + 1)) · ∆X zp k / ∆u squared.

Experimental Parameters

Table I provides experimental parameters and estimated extensive and entangled sizes for mechanical oscillator experiments and proposals. The results show a crucial dependence on temperature, going inversely with ¯n, and on the ratio of zero-point fluctuations of the mode to the atomic fluctuations.

Appendix C Properties

The maximum size theorem states that Nent(ρ, A, Π) = N if and only if ρ has a set of orthogonal eigenstates of the generalised GHZ form. This condition is expressed by Eqs. (C11a) and (C11b). The only maximum-size states for qubits are the pure GHZ-like states √ 1 − q 0⟩ ⊗N + √q e iϕ 1⟩ ⊗N.

Appendix D Oscillator Details

The expansions of local position and momentum operators into normal modes are given by Eq. (D2). The mode functions have the orthogonality relations defined in Eq. (D1). The derivation of the single-particle variance under partitioning leads to Next(ρ, Pk, Π) ∼ 1 / (2¯n + 1) · 1 / Nk · ∆u / ∆X zp k squared.

Appendix E Diffraction Details

The probability R(s) of a diffracted particle passing through the final grating is related to the distribution p(x) by Eq. (E1). The classical Fisher information can be lower-bounded from detection statistics by Eq. (E6). This leads to an extensive size Next(ρ0, Q) ≳ 1.4 × 10 14 for the diffracted molecules. The entangled size estimate Nent(ρ0, Q, Π) ≈ 5 atoms was found in this context.

Concluding Remarks

The paper summarizes the introduction of two measures that capture different ways a systems can display large-scale quantum behaviour.

Improvements for AI systems

Scalable Quantum Macrostate Estimation: Implement a method to estimate extensive size Next(rho, A) using experimental data from micro-mechanical oscillators or diffracting molecules to quantify quantum coherence in position or momentum observables relative to atomic-scale units. This allows AI systems to assess the degree of coherence in large mechanical systems beyond simple decoherence metrics.
Multipartite Entanglement Quantification: Develop a mechanism to calculate entangled size Nent(rho, A, Pi) by quantifying the number of entangled subsystems, which is defined as F(rho, A) divided by the sum over all variances of local observables A Pi acting on each subsystem in a partition Pi. This enables AI to determine if a state exhibits genuine multipartite entanglement, potentially finding evidence for genuine multipartite entanglement of 106 to 107 atoms in recent mechanical superposition states.
Decoherence Rate Modeling: Integrate the relation between decoherence rate Gamma and extensive size Next(rho, Q) into predictive models. Specifically, use the relationship that a typical decoherence rate is related to Next(rho, Q) via Gamma = sigma squared q m2 ua squared 0 tauem2 eħ2 Next(rho, Q), allowing AI to estimate the coherence time tauem based on measured coherence times.
Phase Space State Characterization: Apply Wigner function analysis to mechanical systems where the Quantum Fisher Information (QFI) is known, enabling the calculation of both extensive and entangled sizes from phase space data. This allows AI to characterize complex quantum states, such as a cat state, by finding the QFI-maximizing direction in phase space.
System-Specific Entanglement Witness Generation: Design an entanglement witness Nent(rho, A, Pi) that can be optimized for a specific partition Pi to maximize its value, as shown in Appendix C where the maximum size is achieved when the local operators Ai are constant on the support of rho. This helps AI identify the most relevant subsystems for detecting entanglement in different physical setups.